Verified Answer :
- We factorize the polynomial \(42xy - 49x + 30y - 35\) by grouping the terms as \((42xy - 49x) + (30y - 35)\).
- Extracting the common factor \(7x\) from the first group and \(5\) from the second group results in \(7x(6y - 7) + 5(6y - 7)\).
- Factoring out the shared binomial \((6y - 7)\) yields the final form \((7x + 5)(6y - 7)\).
- The true statements are that the factors are \((7x + 5)\) and \((6y - 7)\). \(\boxed{\text{One of the factors is } (6y - 7), \text{ One of the factors is } (7x + 5)}\).
Explanation
- Grouping Terms
The expression to analyze is \(42xy - 49x + 30y - 35\). To factorize this polynomial, we can use the technique of grouping. This involves grouping the terms in a way that allows us to factor out a common binomial from each pair. We start by arranging the four terms into two groups: \((42xy - 49x)\) and \((30y - 35)\).
- Extracting Common Factors
Now, we extract the greatest common factor from each group.
In the first group, \((42xy - 49x)\), both terms share a common factor of \(7x\). Factoring it out gives \(7x(6y - 7)\).
In the second group, \((30y - 35)\), both terms share a common factor of \(5\). Factoring it out gives \(5(6y - 7)\).
The original expression can now be written as:
\[7x(6y - 7) + 5(6y - 7)\]
- Final Factorization
We can now factor out the common binomial factor \((6y - 7)\) from both terms in the expression:
\[(7x + 5)(6y - 7)\]
Thus, the expression \(42xy - 49x + 30y - 35\) factorizes into \((7x + 5)(6y - 7)\).
- Conclusion and Verification
We compare our findings to the given options:
- One of the factors is \((6y - 7)\). (This is true)
- One of the factors is \((6y + 7)\). (This is false)
- This expression is prime. (This is false)
- One of the factors is \((7x + 5)\). (This is true)
- One of the factors is \((7x - 5)\). (This is false)
The two true statements are that one of the factors is \((6y - 7)\) and one of the factors is \((7x + 5)\).
Examples
Factoring polynomials by grouping is a fundamental tool in algebra, useful for simplifying complex expressions and solving polynomial equations. In practical fields like engineering or physics, this skill helps in solving equations related to dimensions, forces, or wave patterns, where an expression must be broken down to find its roots. For instance, if you are designing a rectangular area where the area is defined by \(42xy - 49x + 30y - 35\), factoring it allows you to identify the length and width as \((7x + 5)\) and \((6y - 7)\) directly.